Abstract

We describe the effect of physical dissipation on stability of
equilibria which have been stabilized, in the absence of damping,
using the method of controlled Lagrangians. This method
applies to a class of underactuated mechanical systems including
“balance” systems such as the pendulum on a cart. Since
the method involves modifying a system’s kinetic energy metric
through feedback, the effect of dissipation is obscured.
In particular, it is not generally true that damping makes a
feedback-stabilized equilibrium asymptotically stable. Damping
in the unactuated directions does tend to enhance stability,
however damping in the controlled directions must be “reversed”
through feedback. In this paper, we suggest a choice
of feedback dissipation to locally exponentially stabilize a class
of controlled Lagrangian systems.